Minimal surfaces of rotation in Finsler space with a Randers metric

 We consider Finsler spaces with a Randers metric F=α+β, on the three dimensional real vector space, where α is the Euclidean metric and β=bdx ₃ is a 1-form with norm b,0≤b<1. By using the notion of mean curvature for immersions in Finsler spaces introduced by Z. Shen, we get the ordinary differential equation that characterizes the minimal surfaces of rotation around the x ₃ axis. We prove that for every b,0≤b<1, there exists, up to homothety, a unique forward complete minimal surface of rotation. The surface is embedded, symmetric with respect to a plane perpendicular to the rotation axis and it is generated by a concave plane curve. Moreover, for every there are non complete minimal surfaces of rotation, which include explicit minimal cones. Received: 30 November 2001 / Published online: 10 February 2003 RID="⋆" ID="⋆" Partially supported by CAPES RID="⋆⋆" ID="⋆⋆" Partially supported by CNPq and PROCAD.     

Kernel estimation for stationary density of Markov chains with general state space

Let {X _n } _n ≥0 be a Markov chain with stationary distributionf(x)ν(dx), ν being a σ-finite measure onE⊂R ^d . Under strict stationarity and mixing conditions we obtain the consistency and asymptotic normality for a general class of kernel estimates off(·). When the assumption of stationarity is dropped these results are extended to geometrically ergodic chains. Partially supported by CAPES. Partially supported by CNPq, PROCAD/CAPES, PRONEX/FAPDF and FINATEC/UnB.     

Quantum Diffusion for the Anderson Model in the Scaling Limit

We consider random Schr?dinger equations on for d ≥ 3 with identically distributed random potential. Denote by λ the coupling constant and ψ_t the solution with initial data ψ₀. The space and time variables scale as with 0 < κ < κ₀(d). We prove that, in the limit λ → 0, the expectation of the Wigner distribution of ψ_t converges weakly to a solution of a heat equation in the space variable x for arbitrary L ² initial data. The diffusion coefficient is uniquely determined by the kinetic energy associated to the momentum υ. This work is an extension to the lattice case of our previous result in the continuum [8,9]. Due to the non-convexity of the level surfaces of the dispersion relation, the estimates of several Feynman graphs are more involved. Submitted: April 18, 2006. Accepted: October 12, 2006. László Erdős: Partially supported by NSF grant DMS-0200235 and EU-IHP Network ‘Analysis and Quantum’ HPRN-CT-2002-0027. Manfred Salmhofer: Partially supported by DFG grant Sa 1362/1-1 and an ESI senior research fellowship. Horng-Tzer Yau: Partially supported by NSF grant DMS-0307295 and MacArthur Fellowship.     

On thep-rank of Ext

AssumeV =L andλ is regular smaller than the first weakly compact cardinal. Under those circumstances and with arbitrary requirements on the structure of Ext (G, ℤ) (under well known limitations), we construct an abelian groupG of cardinalityλ such that for noG′ ⊆G, |G′| <λ isG/G′ free and Ext (G, ℤ) realizes our requirements. Deceased. Partially supported by NSERC. Partially supported by the United States — Israel Binational Science Foundation. Publication No. 314.     

Variation of the conformal radius

We study the change of the conformal radiusr(U) of a simply connected planar domainU versus the subdomainU _ε consisting of the points of distance at least ε to ∂U. We show that the smallest exponent λ such thatr(U)-r(U _t)=0(e ^λ) satisfies 0.59<λ<0.91. We also show that a well-known conjecture implies . Partially supported by NSF Grant DMS-9970398.     

Periodic parabolic problems with concave and convex nonlinearities

Let be a smooth bounded domain, let a, b be two functions that are possibly discontinuous and unbounded with a ≥ 0 in and b > 0 in a set of positive measure and let 0 < p < 1 < q. We prove that there exists some 0 < Λ < ∞ such that the nonlinear Dirichlet periodic parabolic problem in has a positive solution for all 0 < λ < Λ and that there is no positive solution if λ > Λ. In some cases we also show the existence of a minimal solution for all 0 < λ < Λ and that the solution u _λ can be chosen such that λ → u _λ is differentiable and increasing. We also give some upper and lower estimates for such a Λ. All results remain true for the analogous elliptic problems. Partially supported by CONICET, Secyt-UNC, ANPCYT and Agencia Cordoba Ciencia     

Pointwise ergodic theorems for the symmetric exclusion process

Summary Almost sure convergence theorems are proved for Cesaro averages of continous functions in the case of the symmetric exclsion processes in dimension d≧3. For the occupation time of a single site the same result is proved in all dimensions. Partially supported by CNPq     

The Hausdorff Centred measure of the symmetry Cantor sets

Let 0＜λ≤1/3,K (λ) be the attractor of an iterated function system { ψ1,ψ2 } on the line, where ψ1 (x ) =λx, ψ2(x)=1-λ+λx, x∈ [0,1]. We call K (λ) the symmetry Cantor sets. In this paper, we obtained the exact Hausdorff Centred measure of K (λ).     

Further cardinal arithmetic

We continue the investigations in the author’s book on cardinal arithmetic, assuming some knowledge of it. We deal with the cofinality of (S _{≤ℵ 0}(κ), ⊆) for κ real valued measurable (Section 3), densities of box products (Section 5,3), prove the equality cov(λ,λ,θ⁺,2) in more cases even when cf(λ)=ℵ₀ (Section 1), deal with bounds of pp(λ) for λ limit of inaccessible (Section 4) and give proofs to various claims I was sure I had already written but did not find (Section 6). Done mainly 1–4/1991. I thank Alice Leonhardt for typing and retyping so beautifully and accurately. Partially supported by the Basic Research Fund, Israel Academy of Sciences. Publication number 430.     

Almost everywhere convergence of the inverse spherical transform onSL(2,R)

We prove the almost everywhere convergence of the inverse spherical transform ofL _p bi-K-invariant functions on the groupSL(2,R), 4/3<p≤2. The result appears to be sharp. Partially supported by the MPI Partially supported by the CMA     

Weak type estimates for maximal operators with a cylindric distance function

We establish sharp weak-type estimates for the maximal operators T^λ_* associated with cylindric Riesz means for functions on H^p(ℝ³) when 4/5 <p<1 and λ=3/p−5/2, and when p=4/5 and λ>3/p−5/2. The first author was supported by the Korean Research Foundation Grant funded by the Korean Government (MOEHRD) No. R04-2002-000-20028-0. The third author was supported by a Korea University Grant.     

Character formulae for classical groups

We give formulae relating the value X_λ (g) of an irreducible character of a classical group G to entries of powers of the matrix g ε G. This yields a far-reaching generalization of a result of J.L. Cisneros-Molina concerning the GL ₂ case [1]. Partially supported by OTKA grants T 042769 and T 046365     

Spectrum,harmonic functions,and hyperbolic metric spaces

The main result of the paper says, in particular, that ifM is a complete simply connected Riemannian manifold with Ricci curvature bounded from below and without focal points, which is also a hyperbolic metric space in the sense of Gromov, then the top λ of theL ²-spectrum of the Laplace-Beltrami operator Δ is negative, the Martin boundary ofM corresponding to Δ is homeomorphic to the sphere at infinityS(∞), and the harmonic measures onS(∞) have positive Hausdorff dimensions. These generalize the results of [AS], [An1], [Ki], [KL] and [BK]. Moreover, if dimM=2, then in the presence of the other conditions the hyperbolicity is also necessary for λ<0. The machinery consists of a combination of geometrical and probabilistic means. Partially supported by U.S.-Israel BSF. Partially sponsored by the Edmund Landau Center for Research in Mathematical Analysis, supported by the Minerva Foundation (Germany).     

On the Riesz transform associated with the ultraspherical polynomials

We define and investigate the Riesz transform associated with the differential operatorL _λ f(θ)=−f"(θ)−2λ cot’θ. We prove that it can be defined as a principal value and that it is bounded onL ^P ([0, π],dm _λ (θ)),dm _λ(θ)=sin^2λ θdθ, for every 1<p<∞ and of weak type (1,1). The same boundedness properties hold for the maximal operator of the truncated operators. The speed of convergence of the truncated operators is measured in terms of the boundedness inL ^P (dm _λ ), 1<p<∞, and weak type (1,1) of the oscillation and ρ-variation associated to them. Also, a multiplier theorem is proved to get the boundedness of the conjugate function studied by Muckenhoupt and Stein for 1<p<∞ as a corollary of the results for the Riesz transform. Moreover, we find a condition on the weightv which is necessary and sufficient for the existence of a weightu such that the Riesz transform is bounded fromL ^P (v dm _λ ) intoL ^P (u dm _λ ). The authors were partially supported by RTN Harmonic Analysis and Related Problems contract HPRN-CT-2001-00273-HARP. The first and fourth authors were supported in part by KBN grant 1-P93A 018 26. The second and third authors were partially supported by BFM grant 2002-04013-C02-02.     

Completeness of the system {z^λn} in L^2[B,α0]

Let B be an unbounded domain located outside an angle domain with vertex at the origin, A ={λn}（n = 1,2,...） be a sequence of complex numbers satisfying sup ｜ arg（λn）｜ 〈 α 〈 π/2 and denote by M（∧） = {z^λ, λ ∈ ∧} the corresponding system of functions z^λ（λ∈∧）. Let α0（z） be a weight function defined on B. We obtain a completeness theorem for the system M（∧） in the Hilbert space L^2 [B, α0].     

Graded central polynomials for the algebra <Emphasis Type="Italic">M</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(<Emphasis Type="Italic">K</Emphasis>)

Let M _n (K) be the algebra of all n × n matrices over an infinite field K. This algebra has a natural ℤ _n -grading and a natural ℤ-grading. Finite bases for its ℤ _n -graded identities and for its ℤ-graded identities are known. In this paper we describe finite generating sets for the ℤ _n -graded and for the ℤ-graded central polynomials for M _n (K) Partially supported by CNPq 620025/2006-9     

Singular measures andl p Fourier transforms

LetG be the Cantor group or the circle group, and let Γ be the dual ofG. There exists a probability measure λ onG, singular with respect to Haar measure onG, such that for 1≦p<2 the inequality {fx115-1} holds for trigonometric polynomialsf. Partially supported by NSF Grant MCS 76-02267-A01.     

Categoricity of an abstract elementary class in two successive cardinals

We investigate categoricity of abstract elementary classes without any remnants of compactness (like non-definability of well ordering, existence of E.M. models, or existence of large cardinals). We prove (assuming a weak version of GCH around λ) that if ℜ is categorical in λ, λ⁺,LS(ℜ) ≤ λ and has intermediate number of models in λ⁺⁺,then ℜ has a model in λ⁺⁺⁺. Partially supported by the United States-Israel Binational Science Foundation. I thank Alice Leonhardt for the excellent typing. Publication No. 576.     

Identities with involution for the matrix algebra of order two in characteristic<Emphasis Type="Italic">p</Emphasis>

LetM ₂ (K) be the matrix algebra of order two over an infinite fieldK of characteristicp≠2. IfK is algebraically closed then, up to isomorphism, there are two involutions of first kind onM ₂ (K), namely the transpose and the symplectic. IfK is not algebraically closed, studying *-identities it is still sufficient to consider only these two involutions. We describe bases of the polynomial identities with involution in each of these cases. Supported by PhD grant from CNPq. Partially supported by CNPq and by CAPES.     

On ohtsuki’s invariants of integral homology 3-spheres

We provide some more explicit formulae to facilitate the computation of Ohtsuki’s rational invariants λ _n of integral homology 3-spheres extracted from Reshetikhin-TuraevSU(2) quantum invariants. Several interesting consequences will follow from our computation of λ₂. One of them says that λ₂ is always an integer divisible by 3. It seems interesting to compare this result with the fact shown by Murakami that λ₁ is 6 times the Casson invariant. Other consequences include some general criteria for distinguishing homology 3-spheres obtained from surgery on knots by using the Jones polynomial. The first author is supported in part by NSF and the second author is supported by an NSF Postdoctoral Fellowship.